Our analysis proceeds by computing the transition amplitude of simulation error within the \(\eta \) -electron manifold. To this end, we develop multiple techniques for bounding the transition amplitude/expectation of a general fermionic operator, which may be of independent interest. For an \(n\) -spin-orbital electronic-structure problem in the plane-wave basis, our result improves the best previous result in second quantization [1]}, [2]}, [3]} up to a negligible factor while outperforming the first-quantized result [4]} when \(n=\eta ^{2-o(1)}\) . We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.
